A008310 -id:A008310 - cp's OEIS frontend

A063496 a(n) = (2n - 1)(8n^2 - 8n + 3)/3.

1, 19, 85, 231, 489, 891, 1469, 2255, 3281, 4579, 6181, 8119, 10425, 13131, 16269, 19871, 23969, 28595, 33781, 39559, 45961, 53019, 60765, 69231, 78449, 88451, 99269, 110935, 123481, 136939, 151341, 166719, 183105, 200531, 219029, 238631, 259369, 281275, 304381

Offset: 1

N. J. A. Sloane, Aug 01 2001

Number of potential flows in a 2 X 2 matrix with integer velocities in -n..n, i.e., number of 2 X 2 matrices with adjacent elements differing by no more than n, counting matrices differing by a constant only once. - R. H. Hardin, Feb 27 2002
Number of ordered quadruples (a,b,c,d), -(n-1) <= a,b,c,d <= n-1, such that a+b+c+d = 0. - Benoit Cloitre, Jun 14 2003
If Y and Z are 2-blocks of a (2n+1)-set X then a(n-1) is the number of 5-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 28 2007
Equals binomial transform of [1, 18, 48, 32, 0, 0, 0, ...]. - Gary W. Adamson, Jul 19 2008

Harry J. Smith, Table of n, a(n) for n = 1..1000
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
Milan Janjic, Two Enumerative Functions
T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (10).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

(1/12)*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

Cf. A003215, A010006, A142992, A142993, A142994.

[(2*n-1)*(8*n^2-8*n+3)/3: n in [1..40]]; // Wesley Ivan Hurt, May 09 2014

A063496:=n->(2*n-1)*(8*n^2-8*n+3)/3; seq(A063496(n), n=1..40); # Wesley Ivan Hurt, May 09 2014

Table[(2*n - 1)*(8*n^2 - 8*n + 3)/3, {n, 40}] (* Wesley Ivan Hurt, May 09 2014 *)
LinearRecurrence[{4,-6,4,-1}, {1,19,85,231}, 30] (* G. C. Greubel, Dec 01 2017 *)

a(n) = { (2*n - 1)*(8*n^2 - 8*n + 3)/3 } \\ Harry J. Smith, Aug 23 2009

my(x='x+O('x^30)); Vec(serlaplace((-3+6*x+24*x^2+16*x^3)*exp(x)/3 + 1)) \\ G. C. Greubel, Dec 01 2017

From Peter Bala, Jul 18 2008: (Start)
The following remarks about the C_3 lattice assume the sequence offset is 0.
Partial sums of A010006. So this sequence is the crystal ball sequence for the C_3 lattice - row 3 of A142992. The lattice C_3 consists of all integer lattice points v = (a,b,c) in Z^3 such that a + b + c is even, equipped with the taxicab type norm ||v|| = (1/2) * (|a| + |b| + |c|).
The crystal ball sequence of C_3 gives the number of lattice points v in C_3 with ||v|| <= n for n = 0,1,2,3,... [Bacher et al.].
For example, a(1) = 19 because the origin has norm 0 and the 18 lattice points in Z^3 of norm 1 (as defined above) are +-(2,0,0), +-(0,2,0), +-(0,0,2), +-(1,1,0), +-(1,0,1), +-(0,1,1), +-(1,-1,0), +-(1,0,-1) and +-(0,1,-1). These 18 vectors form a root system of type C_3.
O.g.f.: x*(1 + 15*x + 15*x^2 + x^3)/(1 - x)^4 = x/(1 - x) * T(3, (1 + x)/(1 - x)), where T(n, x) denotes the Chebyshev polynomial of the first kind.
2*log(2) = 4/3 + Sum_{n >= 1} 1/(n*a(n)*a(n+1)). (End)
a(n+1) = (1/Pi) * Integral_{x=0..Pi} (sin((n+1/2)*x)/sin(x/2))^4. - Yalcin Aktar, Nov 02 2011, corrected by R. J. Mathar, Dec 01 2011
From G. C. Greubel, Dec 01 2017: (Start)
G.f.: x*(1 + 15*x + 15*x^2 + x^3)/(1 - x)^4.
E.g.f.: (-3 + 6*x + 24*x^2 + 16*x^3)*exp(x)/3 + 1. (End)
a(n) = A005900(2n-1). - Ivan N. Ianakiev, Mar 27 2022
From Peter Bala, Mar 11 2024: (Start)
Sum_{k = 1..n+1} 1/(k*a(k)*a(k+1)) = 1/(19 - 3/(27 - 60/(43 - 315/(67 - ... -n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*3^2))))).
E.g.f.: exp(x)*(1 + 18*x + 48*x^2/2! + 32*x^3/3!). Note that -T(6, i*sqrt(x)) = 1 + 18*x + 48*x^2 + 32*x^3, where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. See A008310. (End)

A028297 Coefficients of Chebyshev polynomials of the first kind: triangle of coefficients in expansion of cos(n*x) in descending powers of cos(x).

Original entry on oeis.org

1, 1, 2, -1, 4, -3, 8, -8, 1, 16, -20, 5, 32, -48, 18, -1, 64, -112, 56, -7, 128, -256, 160, -32, 1, 256, -576, 432, -120, 9, 512, -1280, 1120, -400, 50, -1, 1024, -2816, 2816, -1232, 220, -11, 2048, -6144, 6912, -3584, 840, -72, 1, 4096, -13312, 16640, -9984

Offset: 0

N. J. A. Sloane

Rows are of lengths 1, 1, 2, 2, 3, 3, ... (A008619).
This triangle is generated from A118800 by shifting down columns to allow for (1, 1, 2, 2, 3, 3, ...) terms in each row. - Gary W. Adamson, Dec 16 2007
Unsigned triangle = A034839 * A007318. - Gary W. Adamson, Nov 28 2008
Triangle, with zeros omitted, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, -1, 1, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 16 2011
From Wolfdieter Lang, Aug 02 2014: (Start)
This irregular triangle is the row reversed version of the Chebyshev T-triangle A053120 given by A039991 with vanishing odd-indexed columns removed.
If zeros are appended in each row n >= 1, in order to obtain a regular triangle (see the Philippe Deléham comment, g.f. and example) this becomes the Riordan triangle (1-x)/(1-2*x), -x^2/(1-2*x). See also the unsigned version A201701 of this regular triangle.
(End)
Apparently, unsigned diagonals of this array are rows of A200139. - Tom Copeland, Oct 11 2014
It appears that the coefficients are generated by the following: Let SM_k = Sum( d_(t_1, t_2)* eM_1^t_1 * eM_2^t_2) summed over all length 2 integer partitions of k, i.e., 1*t_1 + 2*t_2 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 2 data (i.e., SM_k = S_k/2 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(2,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2) form an irregular triangle, with one row for each k value starting with k=1. Thus this procedure and associated OEIS sequences A287768, A288199, A288207, A288211, A288245, A288188 are generalizations of Chebyshev polynomials of the first kind. - Gregory Gerard Wojnar, Jul 01 2017

			Letting c = cos x, we have: cos 0x = 1, cos 1x = 1c; cos 2x = 2c^2-1; cos 3x = 4c^3-3c, cos 4x = 8c^4-8c^2+1, etc.
T4 = 8x^4 - 8x^2 + 1 = 8, -8, +1 = 2^(3) - (4)(2) + [2^(-1)](4)/2.
From _Wolfdieter Lang_, Aug 02 2014: (Start)
The irregular triangle T(n,k) begins:
n\k     1      2     3      4     5     6   7   8 ....
0:      1
1:      1
2:      2     -1
3:      4     -3
4:      8     -8     1
5:     16    -20     5
6:     32    -48    18     -1
7:     64   -112    56     -7
8:    128   -256   160    -32     1
9:    256   -576   432   -120     9
10:   512  -1280  1120   -400    50    -1
11:  1024  -2816  2816  -1232   220   -11
12:  2048  -6144  6912  -3584   840   -72   1
13:  4096 -13312 16640  -9984  2912  -364  13
14:  8192 -28672 39424 -26880  9408 -1568  98  -1
15: 16384 -61440 92160 -70400 28800 -6048 560 -15
...
T(4,x) = 8*x^4 -8*x^2 + 1*x^0, T(5,x) = 16*x^5 - 20*x^3 + 5*x^1, with Chebyshev's T-polynomials (A053120). (End)
From _Philippe Deléham_, Dec 16 2011: (Start)
The triangle (1,1,0,0,0,0,...) DELTA (0,-1,1,0,0,0,0,...) includes zeros and begins:
   1;
   1,   0;
   2,  -1,  0;
   4,  -3,  0,  0;
   8,  -8,  1,  0, 0;
  16, -20,  5,  0, 0, 0;
  32, -48, 18, -1, 0, 0, 0; (End)

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed., Section 1.335, p. 35.
S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 106. [From Rick L. Shepherd, Jul 06 2010]

Alois P. Heinz, Rows n = 0..200, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy] p. 795.
Pantelis A. Damianou, A Beautiful Sine Formula, Amer. Math. Monthly 121 (2014), no. 2, 120--135. MR3149030.
Daniel J. Greenhoe, Frames and Bases: Structure and Design, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, see page 172.
Daniel J. Greenhoe, A Book Concerning Transforms, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 94.
Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, Table GW.n=2, p. 22, arXiv:1706.08381 [math.GM], 2017.

Cf. A028298.
Reflection of A008310, the main entry. With zeros: A039991.
Cf. A053120 (row reversed table including zeros).
Cf. A118800, A034839, A081277, A124182.
Cf. A001333 (row sums 1), A001333 (alternating row sums). - Wolfdieter Lang, Aug 02 2014

b:= proc(n) b(n):= `if`(n<2, 1, expand(2*b(n-1)-x*b(n-2))) end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..15);  # Alois P. Heinz, Sep 04 2019

t[n_] := (Cos[n x] // TrigExpand) /. Sin[x]^m_ /; EvenQ[m] -> (1 - Cos[x]^2)^(m/2) // Expand; Flatten[Table[ r = Reverse @ CoefficientList[t[n], Cos[x]]; If[OddQ[Length[r]], AppendTo[r,0]]; Partition[r,2][[All, 1]],{n, 0, 13}] ][[1 ;; 53]] (* Jean-François Alcover, May 06 2011 *)
Tpoly[n_] := HypergeometricPFQ[{(1 - n)/2, -n/2}, {1/2}, 1 - x];
Table[CoefficientList[Tpoly[n], x], {n, 0, 12}] // Flatten (* Peter Luschny, Feb 03 2021 *)

cos(n*x) = 2 * cos((n-1)*x) * cos(x) - cos((n-2)*x) (from CRC's Multiple-angle relations). - Rick L. Shepherd, Jul 06 2010
G.f.: (1-x) / (1-2x+y*x^2). - Philippe Deléham, Dec 16 2011
Sum_{k=0..n} T(n,k)*x^k = A011782(n), A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x = 0, 1, 2, 3, 4, 5, 6, respectively. - Philippe Deléham, Dec 16 2011
T(n,k) = [x^k] hypergeom([1/2 - n/2, -n/2], [1/2], 1 - x). - Peter Luschny, Feb 03 2021
T(n,k) = (-1)^k * 2^(n-1-2*k) * A034807(n,k). - Hoang Xuan Thanh, Jun 21 2025

More terms from David W. Wilson

Row length sequence and link to Abramowitz-Stegun added by Wolfdieter Lang, Aug 02 2014

A001794 Negated coefficients of Chebyshev T polynomials: [x^n](-T(n+6, x)), n >= 0.

Original entry on oeis.org

1, 7, 32, 120, 400, 1232, 3584, 9984, 26880, 70400, 180224, 452608, 1118208, 2723840, 6553600, 15597568, 36765696, 85917696, 199229440, 458752000, 1049624576, 2387607552, 5402263552, 12163481600, 27262976000, 60850962432

Offset: 0

N. J. A. Sloane

A negated subdiagonal of A053120.
If X_1,X_2,...,X_n are 2-blocks of a (2n+1)-set X then a(n-2) is the number of (n+3)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007
The third corrector line for transforming 2^n offset 0 with a leading 1 into the Fibonacci sequence. - Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009
a(n-2) is the number of strings of length n defined on {0, 1, 2, 3} that have exactly two 2's and no 3's or exactly three 3's and no 2's. For example, for n=4, a(2)=32 since the strings are the 6 permutations of 2200, the 12 permutations of 2201, the 6 permutations of 2211, the 4 permutations of 3330, and the 4 permutations of 3331. - Enrique Navarrete, Jun 03 2025

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Michael Albert, Mike Atkinson and Robert Brignall, The enumeration of three pattern classes using monotone grid classes, E. J. Combinat., Vol. 19, No. 3 (2012), Article P20, chapter 5.4.1.
Takayuki Hibi, Nan Li and Hidefumi Ohsugi, The face vector of a half-open hypersimplex, arXiv preprint arXiv:1309.5155 [math.CO], 2013-2014.
Milan Janjic, Two Enumerative Functions.
C. W. Jones, J. C. P. Miller, J. F. C. Conn, and R. C. Pankhurst, Tables of Chebyshev polynomials, Proc. Roy. Soc. Edinburgh. Sect. A., Vol. 62, No. 2 (1946), pp. 187-203.
John Machacek, A finite totally nonnegative Grassmannian, arXiv:2410.06177 [math.CO], 2024. See p. 11.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A039991 (negative of column 6), A028297, A008310, A053120.
With alternating signs, the o.g.f. (with offset 1) is the inverse of the o.g.f. of A065097.
Cf. A001789 (partial sums), A081279 (binomial transform), A005900 (0 followed by inverse binomial transform).

List([0..25],n->2^(n-2)*(n+1)*(n+2)*(n+6)/3); # Muniru A Asiru, Mar 20 2018

[2^(n-1)/3*Binomial(n+2,2)*(n+6) : n in [0..25]]; // Brad Clardy, Mar 08 2012

[seq(coeftayl((1-x)/(1-2*x)^4, x = 0, k), k=0..25)]; # Muniru A Asiru, Mar 20 2018

a[n_] := 2^(n-2)*(n+1)*(n+2)*(n+6)/3; a /@ Range[0, 20] (* Giovanni Resta, Mar 25 2017 *)
LinearRecurrence[{8,-24,32,-16},{1,7,32,120},30] (* Harvey P. Dale, Oct 08 2024 *)

a(n) = sum(i=0, n+1, sum(k=0, i, k^2*binomial(n+1, i))); \\ Michel Marcus, Mar 25 2017

a(n) = - polcoeff(polchebyshev(n+6), n); \\ Michel Marcus, Mar 20 2018

a(n) = 2^(n-2)*(n+1)*(n+2)*(n+6)/3. [See a comment in A053120 on subdiagonals. - Wolfdieter Lang, Jan 03 2020]
G.f.: (1-x)/(1-2*x)^4. - Simon Plouffe in his 1992 dissertation
a(n) = Sum_{k=0..floor((n+6)/2)} C(n+6, 2*k)*C(k, 3). - Paul Barry, May 15 2003
With a leading zero, the binomial transform of A000330. - Paul Barry, Jul 19 2003
a(n) = Sum_{i=0..n+1} (Sum{k=0..i} (k^2*binomial(n+1, i))). - Jon Perry, Feb 26 2004 [corrected by Michel Marcus, Mar 25 2017]
Binomial transform of a(n) = (2*n^3 + 6*n^2 + 7*n + 3)/3 offset 0. a(3)=120. - Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009
a(n) = (2^(n-1)/3)*binomial(n+2,2)*(n+6). - Brad Clardy, Mar 08 2012
E.g.f.: (1/3)*exp(2*x)*(3 + 15*x + 12*x^2 + 2*x^3). - Stefano Spezia, Jan 03 2020
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=0} 1/a(n) = 156*log(2)/5 - 511/25.
Sum_{n>=0} (-1)^n/a(n) = 241/25 - 108*log(3/2)/5. (End)
E.g.f.: exp(2*x)*(x^2/2 + x^3/6) (with two leading zeros). - Enrique Navarrete, Jun 03 2025

Name clarified by Wolfdieter Lang, Nov 26 2019

A039991 Triangle of coefficients of cos(x)^n in polynomial for cos(nx).

Original entry on oeis.org

1, 1, 0, 2, 0, -1, 4, 0, -3, 0, 8, 0, -8, 0, 1, 16, 0, -20, 0, 5, 0, 32, 0, -48, 0, 18, 0, -1, 64, 0, -112, 0, 56, 0, -7, 0, 128, 0, -256, 0, 160, 0, -32, 0, 1, 256, 0, -576, 0, 432, 0, -120, 0, 9, 0, 512, 0, -1280, 0, 1120, 0, -400, 0, 50, 0, -1, 1024, 0, -2816, 0, 2816, 0, -1232, 0, 220, 0, -11, 0

Offset: 0

David W. Wilson

Also triangle of coefficients of Chebyshev polynomials of first kind (T(n,x)) in decreasing order of powers of x. A053120 gives the coefficients in increasing order.

The polynomials R(n,x) := Sum_{m=0..n} a(n,m)*sqrt(x)^m, have g.f. (1-z)/(1 - 2*z + x*z^2) = ((1-z)/(1-2*z))/(1 - x*(-z^2/(1-2*z))) (from the row reversion of the g.f. of A053120 and x^2 -> x). Therefore this triangle becomes the Riordan triangle ((1-z)/(1-2*z), -z^2/(1-2*z)) if the vanishing columns are deleted (see A028297) and zeros are appended in each row numbered n>=1 in order to obtain a triangle. This is then A201701 with negative odd numbered columns. - Wolfdieter Lang, Aug 06 2014

			Letting c = cos x, we have: cos 0x = 1, cos 1x = 1c; cos 2x = 2c^2-1; cos 3x = 4c^3-3c, cos 4x = 8c^4-8c^2+1, etc.
From _Wolfdieter Lang_, Aug 06 2014: (Start)
The triangle a(n,m) begins:
  n\m    0 1     2 3    4 5     6 7   8 9  10 11 ...
  0:     1
  1:     1 0
  2:     2 0    -1
  3:     4 0    -3 0
  4:     8 0    -8 0    1
  5:    16 0   -20 0    5 0
  6:    32 0   -48 0   18 0    -1
  7:    64 0  -112 0   56 0    -7 0
  8:   128 0  -256 0  160 0   -32 0   1
  9:   256 0  -576 0  432 0  -120 0   9 0
  10:  512 0 -1280 0 1120 0  -400 0  50 0  -1
  11: 1024 0 -2816 0 2816 0 -1232 0 220 0 -11  0
  ...
--------------------------------------------------------------------------
Chebyshev T-polynomials (decreasing even or odd powers):
n=3: T(3, x) = 4*x^3 - 3*x^1; n=4: T(4, x) = 8*x^4 - 8*x^2 + 1. (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
Martin Aigner and Gunter M. Ziegler, Proofs From the Book, Springer 2004. See Chapter 18, Appendix.
E. A. Guilleman, Synthesis of Passive Networks, Wiley, 1957, p. 593.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

T. D. Noe, Table of n, a(n) for n = 0..5150
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Daniel J. Greenhoe, Frames and Bases: Structure and Design, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, see page 172.
Daniel J. Greenhoe, A Book Concerning Transforms, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 94.
Index entries for sequences related to Chebyshev polynomials.

Cf. A028297 (without vanishing columns). A008310 (zero columns deleted then rows reversed).
Triangle without zeros: A028297. Without signs: A081265.
Cf. A053120 (increasing powers of x).

function T(n,k) // T = A039991
  if k lt 0 or k gt n then return 0;
  elif n lt 2 and k eq 0 then return 1;
  else return 2*T(n-1, k) - T(n-2, k-2);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 10 2022

seq(seq(coeff(orthopoly[T](i,x),x,i-j),j=0..i),i=0..20); # Robert Israel, Aug 07 2014

row[n_] := CoefficientList[ ChebyshevT[n, x], x] // Reverse; Table[row[n], {n, 0, 11}] // Flatten(* Jean-François Alcover, Sep 14 2012 *)

T(n,m)=(1+(-1)^m)*(binomial(n-m/2,n-m)+binomial(n-1-m/2,n-m))*2^(n-m-2)*(-1)^((m+1-(-1)^m)/2) /* Tani Akinari, Jul 18 2024 */

def T(n, k): # T = A039991
    if (n<2 and k==0): return 1
    elif (k<0 or k>n): return 0
    else: return 2*T(n-1, k) - T(n-2, k-2)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 10 2022

T(n, m) = 0 if n= 2, m >= 0; T(n, -2) = T(n, -1) = 0, T(0, 0) = T(1, 0) = 1.
G.f. for m-th column: 0 if m odd, (1-x)/(1-2*x) if m=0, else ((-1)^(m/2))*(x^m)*(1-x)/(1-2*x)^(m/2+1). For g.f. for row polynomials and row sums, see A053120.
G.f. row polynomials: (1-z)/(1 - 2*z + (x*z)^2). - Wolfdieter Lang, Aug 06 2014
Recurrence for the row polynomials Trev(n, x):= x^n*T(n, 1/x) = Sum_{m=0..n} T(n, m)*x^m; Trev(n, x) = 2*Trev(n-1, x) - x^2*Trev(n-2, x), n >= 1, Trev(-1, x) = 1/x^2 and Trev(0, x) = 1. From the T(n, x) recurrence. Compare this with A081265. - Wolfdieter Lang, Aug 07 2014
T(n,m) = (1+(-1)^m)*(binomial(n-m/2,n-m)+binomial(n-1-m/2,n-m))*2^(n-m-2)*(-1)^((m+1-(-1)^m)/2). - Tani Akinari, Jul 18 2024

Entry improved by comments from Wolfdieter Lang, Jan 11 2000.

Edited: A053120 added in comment and crossrefs. Cfs. A028297 and A008310 specified. - Wolfdieter Lang, Aug 06 2014

A123027 Triangle of coefficients of (1 - x)^nU(n,-(3x - 2)/(2*x - 2)), where U(n,x) is the n-th Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, -2, 3, 3, -10, 8, -4, 22, -38, 21, 5, -40, 111, -130, 55, -6, 65, -256, 474, -420, 144, 7, -98, 511, -1324, 1836, -1308, 377, -8, 140, -924, 3130, -6020, 6666, -3970, 987, 9, -192, 1554, -6588, 16435, -25088, 23109, -11822, 2584, -10, 255, -2472, 12720, -39430, 77645, -98160, 77378, -34690, 6765

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 24 2006

The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A053122(n,j)*x^j*(1 - x)^(n - j).

			Triangle begins:
     1;
    -2,    3;
     3,  -10,    8;
    -4,   22,   38,    21;
     5,  -40,  111,  -130,    55;
    -6,   65, -256,   474,  -420,    144;
     7,  -98,  511, -1324,  1836,  -1308,   377;
    -8,  140, -924,  3130, -6020,   6666, -3970,    987;
     9, -192, 1554, -6588, 16435, -25088, 23109, -11822, 2584;
     ... reformatted and extended. _Franck Maminirina Ramaharo_, Oct 10 2018

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the Second Kind
Wikipedia, Chebyshev polynomials

Cf. A008310, A049310, A053122, A111006.

Cf. A122753, A123018, A123019, A123021, A123199, A123202, A123217, A123221.

b0 = Table[CoefficientList[ChebyshevU[n, x/2 -1], x], {n, 0, 10}];
Table[CoefficientList[Sum[b0[[m+1]][[n+1]]*x^n*(1-x)^(m-n), {n,0,m}], x], {m,0,10}]//Flatten
(* Alternative Adamson Matrix method *)
t[n_, m_] = If[n==m, 2, If[n==m-1 || n==m+1, 1, 0]];
M[d_] := Table[t[n, m], {n, d}, {m, d}];
a = Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]];
Table[CoefficientList[Sum[a[[m+1]][[n+1]]*x^n*(1-x)^(m-n), {n, 0, m}], x], {m, 0, 10}]//Flatten

A053122(n, k) := if n < k then 0 else ((-1)^(n - k))*binomial(n + k + 1,  2*k + 1)$
P(x, n) := expand(sum(A053122(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$
T(n, k) := ratcoef(P(x, n), x, k)$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, hipow(P(x, n), x)))$ /* Franck Maminirina Ramaharo, Oct 10 2018 */

def A053122(n, k): return 0 if (nA053122(n, j)*x^j*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
flatten([T(n) for n in (0..12)]) # G. C. Greubel, Jul 15 2021

From Franck Maminirina Ramaharo, Oct 10 2018: (Start)
Row n = coefficients in the expansion of (1/sqrt((5*x - 4)*x))*(((3*x - 2 + sqrt((5*x - 4)*x))/2)^(n + 1) - ((3*x - 2 - sqrt((5*x - 4)*x))/2)^(n + 1)).
G.f.: 1/(1 + (2 - 3*x)*t + (1 - x)^2*t^2).
E.g.f.: exp(t*(3*x - 2)/2)*(sqrt((5*x - 4)*x)*cosh(t*sqrt((5*x - 4)*x)/2) + (3*x - 2)*sinh(t*sqrt((5*x - 4)*x)/2))/sqrt((5*x - 4)*x).
T(n,1) = (-1)^(n+1)*A006503(n).
T(n,n) = A001906(n+1). (End)

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 10 2018

A008312 Triangle of coefficients of Chebyshev polynomials U_n(x).

Original entry on oeis.org

1, 2, -1, 4, -4, 8, 1, -12, 16, 6, -32, 32, -1, 24, -80, 64, -8, 80, -192, 128, 1, -40, 240, -448, 256, 10, -160, 672, -1024, 512, -1, 60, -560, 1792, -2304, 1024, -12, 280, -1792, 4608, -5120, 2048

Offset: 0

N. J. A. Sloane

Version with zeros in A053117. - Philippe Deléham, Nov 27 2013

			From _Philippe Deléham_, Nov 27 2013: (Start)
Triangle begins:
    1;
    2;
   -1,    4;
   -4,    8;
    1,  -12,    16;
    6,  -32,    32;
   -1,   24,   -80,    64;
   -8,   80,  -192,   128;
    1,  -40,   240,  -448,   256;
   10, -160,   672, -1024,   512;
   -1,   60,  -560,  1792, -2304, 1024;
  -12,  280, -1792,  4608, -5120, 2048;
  ...
With zeros, triangle begins:
   1;
   0,   2;
  -1,   0,   4;
   0,  -4,   0,    8;
   1,   0, -12,    0,   16;
   0,   6,   0,  -32,    0,    32;
  -1,   0,  24,    0,  -80,     0,   64;
   0,  -8,   0,   80,    0,  -192,    0,   128;
   1,   0, -40,    0,  240,     0, -448,     0,   256;
   0,  10,   0, -160,    0,   672,    0, -1024,     0,   512;
  -1,   0,  60,    0, -560,     0, 1792,     0, -2304,     0, 1024;
   0, -12,   0,  280,    0, -1792,    0,  4608,     0, -5120,    0, 2048;
  ...
(End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.

T. D. Noe, Rows n = 0..100 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
D. Foata and G.-N. Han, Nombres de Fibonacci et polynomes orthogonaux.
Valentin Ovsienko, Towards quantized complex numbers: q-deformed Gaussian integers and the Picard group, arXiv:2103.10800 [math.QA], 2021.
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From _N. J. A. Sloane_, Feb 13 2013
Index entries for sequences related to Chebyshev polynomials.

Reflected version with zeros: A053118. Cf. A008310, A053112, A053117.

a[n_, k_] := Coefficient[ ChebyshevU[n, x], x, k]; row[n_] := Table[a[n, k], {k, Mod[n, 2], n, 2}]; Table[row[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Oct 03 2012 *)

A084930 Triangle of coefficients of Chebyshev polynomials T_{2n+1} (x).

Original entry on oeis.org

1, -3, 4, 5, -20, 16, -7, 56, -112, 64, 9, -120, 432, -576, 256, -11, 220, -1232, 2816, -2816, 1024, 13, -364, 2912, -9984, 16640, -13312, 4096, -15, 560, -6048, 28800, -70400, 92160, -61440, 16384, 17, -816, 11424, -71808, 239360, -452608, 487424, -278528, 65536, -19, 1140, -20064, 160512, -695552

Offset: 0

Gary W. Adamson, Jun 12 2003

From Herb Conn, Jan 28 2005: (Start)
"Letting x = 2 Cos 2A, we have the following trigonometric identities:
"Sin 3A = 3*Sin A - 4*Sin^3 A
"Sin 5A = 5*Sin A - 20*Sin^3 A + 16*Sin^5 A
"Sin 7A = 7*Sin A - 56*Sin^3 A + 112*Sin^5 A - 64*Sin^7 A
"Sin 9A = 9*Sin A - 120*Sin^3 A + 432*Sin^5 A - 576*Sin^7 A + 256*Sin^9 A, etc." (End)
Cayley (1876) states "Write sin u = x, then we have  sin u = x, [...] sin 3u = 3x - 4x^3, [...] sin 5u = 5x - 20x^3 + 16 x^5, [...]". Since T_n(cos(u)) = cos(nu) for all integer n, sin(u) = cos(u - Pi/2), and sin(u + k*Pi) = (-1)^k sin(u) it follows that T_n(sin(u)) = (-1)^((n-1)/2) sin(nu) for all odd integer n. - Michael Somos, Jun 19 2012
From Wolfdieter Lang, Aug 05 2014: (Start)
The coefficient triangle t(n,k) for the row polynomials Todd(n, x) := T_{2*n+1}(sqrt(x))/sqrt(x) = sum(t(n,k)*x^k, k=0..n) is the Riordan triangle ((1-z)/(1+z)^2, 4*z/(1+z)^2) (rewrite the g.f. for the present triangle a(n,k) given in the formula section). The triangle entries t(n,k) = a(n,k), but the interpretation of the row polynomials is different for both cases.
From the relation Todd(n, x) = S(n, 2*(2*x-1)) - S(n-1, 2*(2*x-1)) with the Chebyshev S-polynomials (see A049310 and the formula section of A130777) follows the recurrence: Todd(n, x) = 2*(-1)^n*(1-x)*Todd(n-1, 1-x) + (2x-1)*Todd(n-1, x), n >= 1, Todd(0, x) = 1.
This corresponds to the triangle recurrence t(n,k) = (2*(k+1)*(-1)^(n-k) - 1)*t(n-1,k) + 2*(1 +(-1)^(n-k))*t(n-1,k-1) + 2*(-1)^(n-k)*sum(binomial(l+1,k)*t(n-1,l), l=k+1..n-1), n >= k >= 1, t(n,k) = 0 if n < k, t(n,0) = (-1)^n*(2*n+1). Compare this with the shorter recurrence involving the rational A-sequence for this Riordan triangle which has g.f. x^2/(2-x-2*sqrt(1-x)). t(n,k) = sum(A(j)*t(n-1,k-1+j), j=0..n-k), n >= k >= 1. The Z-sequence has g.f. -(1 + 2/sqrt(1-x)). For the A- and Z-sequence see a link under A006232. (End)

			The triangle a(n,k):
n   2n+1\k 0     1      2       3       4        5        6         7        8        9      10 ...
0    1:    1
1    3:   -3     4
2    5:    5   -20     16
3    7:   -7    56   -112      64
4    9:    9  -120    432    -576     256
5   11:  -11   220  -1232    2816   -2816     1024
6   13:   13  -364   2912   -9984   16640   -13312     4096
7   15:  -15   560  -6048   28800  -70400    92160   -61440     16384
8   17:   17  -816  11424  -71808  239360  -452608   487424   -278528    65536
9   19:  -19  1140 -20064  160512 -695552  1770496 -2723840   2490368 -1245184   262144
10  21:   21 -1540  33264 -329472 1793792 -5870592 12042240 -15597568 12386304 -5505024 1048576
... formatted and extended by _Wolfdieter Lang_, Aug 02 2014.
---------------------------------------------------------------------------------------------------
First few polynomials T_{2n+1}(x) are
1*x - 3*x + 4*x^3
5*x - 20*x^3 + 16*x^5
- 7*x + 56*x^3 - 112*x^5 + 64*x^7
9*x - 120*x^3 + 432*x^5 - 576*x^7  + 256*x^9

A. Cayley, On an Expression for 1 +- sin(2p+1)u in Terms of sin u, Messenger of Mathematics, 5 (1876), pp. 7-8 = Mathematical Papers Vol. 10, n. 630, pp. 1-2.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2nd ed., Wiley, New York, 1990. p. 37, eq. (1.96) and p. 4, eq. (1.10).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 795.
Index entries for sequences related to Chebyshev polynomials.

Cf. A002315, A028297.
Cf. A053120 (coefficient triangle of T-polynomials), A127674 (even-indexed T polynomials).
Cf. A127675 (row reversed triangle with different signs).
Cf. A006232, A008310, A049310, A130777.
Cf. A157142, A000330(n), A005585, A050486, A054333.

row[n_] := (T = ChebyshevT[2*n+1, x]; Coefficient[T, x, #]& /@ Range[1, Exponent[T, x], 2]); Table[row[n], {n, 0, 9} ] // Flatten (* Jean-François Alcover, Aug 06 2014 *)

Alternate rows of A008310.
a(n,k)=((-1)^(n-k))*(2^(2*k))*binomial(n+1+k,2*k+1)*(2*n+1)/(n+1+k) if n>=k>=0 else 0.
From Wolfdieter Lang, Aug 02 2014: (Start)
a(n,k) = [x^(2*k+1)] T_{2*n+1}(x), n >= k >= 0.
G.f. for row polynomials: x*(1-z)/(1 + 2*(1- 2*x^2)*z + z^2). (End)
The first column sequences are: A157142, 4*(-1)^(n+1)*A000330(n), 16*(-1)^n*A005585(n-1), 64*(-1)^(n+1)*A050486(n-3), 256*(-1)^n*A054333(n-4), ... - Wolfdieter Lang, Aug 05 2014

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jun 26 2003

Edited; example rewritten (to conform with the triangle) by Wolfdieter Lang, Aug 02 2014

A053112 Expansion of (-1 + 1/(1-9x)^9)/(81x); related to A053108.

Original entry on oeis.org

1, 45, 1485, 40095, 938223, 19702683, 379980315, 6839645670, 116273976390, 1883638417518, 29282015399598, 439230230993970, 6385731819835410, 90312492880529370, 1246312401751305306, 16825217423642621631

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (81, -2916, 61236, -826686, 7440174, -44641044, 172186884, -387420489, 387420489).

Cf. A053108, A053110 - A053113.

Without signs: A078812. With zeros: A049310. Cf. A008310 (T(n, x)), A008312 (U(n, x)).

[9^(n-1)*Binomial(n+9, 8): n in [0..30]]; // G. C. Greubel, Aug 16 2018

CoefficientList[Series[(-1+1/(1-9*x)^9)/(81*x),{x,0,30}],x] (* or *) LinearRecurrence[{81,-2916,61236,-826686,7440174,-44641044,172186884,-387420489,387420489}, {1,45,1485,40095,938223,19702683, 379980315, 6839645670,116273976390},20] (* Harvey P. Dale, Apr 27 2013 *)
Table[9^(n - 1)*Binomial[n + 9, 8], {n, 0, 30}] (* G. C. Greubel, Aug 16 2018 *)

vector(30,n,n--; 9^(n-1)*binomial(n+9, 8)) \\ G. C. Greubel, Aug 16 2018

G.f.: (-1 + 1/(1-9*x)^9)/(81*x).
a(n) = 9^(n-1)*binomial(n+9, 8).
a(0)=1, a(1)=45, a(2)=1485, a(3)=40095, a(4)=938223, a(5)=19702683, a(6)=379980315, a(7)=6839645670, a(8)=116273976390, a(n)=81*a(n-1)- 2916*a(n-2)+ 61236*a(n-3)- 826686*a(n-4)+ 7440174*a(n-5)- 44641044*a(n-6)+ 172186884*a(n-7)- 387420489*a(n-8)+ 387420489*a(n-9). - Harvey P. Dale, Apr 27 2013

A142993 Crystal ball sequence for the lattice C_4.

Original entry on oeis.org

1, 33, 225, 833, 2241, 4961, 9633, 17025, 28033, 43681, 65121, 93633, 130625, 177633, 236321, 308481, 396033, 501025, 625633, 772161, 943041, 1140833, 1368225, 1628033, 1923201

Offset: 0

Peter Bala, Jul 18 2008

The lattice C_4 consists of all integer lattice points v = (a,b,c,d) in Z^4 such that a + b + c + d is even, equipped with the taxicab type norm ||v|| = (1/2) * (|a| + |b| + |c| + |d|). The crystal ball sequence of C_4 gives the number of lattice points v in C_4 with ||v|| <= n for n = 0,1,2,3,... [Bacher et al.].

			a(1) = 33. The origin has norm 0. The 32 lattice points in Z^4 of norm 1 (as defined above) are +-2*e_i, 1 <= i <= 4 and (+- e_i +- e_j), 1 <= i < j <= 4, where e_1, e_2, e_3 and e_4 denotes the standard basis of Z^4. These 32 vectors form a root system of type C_4. Hence sequence begins 1, 1 + 32 = 33, ... .

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, Annales de l'Institut Fourier, Tome 49 (1999) no. 3, pp. 727-762.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
Peter Bala, The C_4 lattice and a continued fraction for log(2)
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A019560, A063496, A142992, A142994.

a := n -> (2*n+1)^2*(4*n^2+4*n+3)/3: seq(a(n), n = 0..24)

Partial sums of A019560. a(n) = (2*n+1)^2*(4*n^2+4*n+3)/3 = Sum_{k = 0..4} C(8,2k)*C(n+k,4) = Sum_{k = 0..4} C(8,2k+1)*C(n+k+1/2,4). O.g.f.: (1+28*x+70*x^2+28*x^3+x^4)/(1-x)^5 = (1/(1-x)) * T(4,(1+x)/(1-x)), where T(n,x) denotes the Chebyshev polynomial of the first kind. 2*log(2) = 17/12 - Sum_{n >= 1} 1/(n*a(n-1)*a(n)).
From Peter Bala, Mar 11 2024: (Start)
Sum_{k = 1..n+1} 1/(k*a(k)*a(k-1)) = 1/(33 - 3/(41 - 60/(57 - 315/(81 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*4^2))))).
E.g.f.: exp(x)*(1 + 32*x + 160*x^2/2! + 256*x^3/3! + 128*x^4/4!).
Note that T(8, i*sqrt(x)) = 1 + 32*x + 160*x^2 + 256*x^3 + 128*x^4. See A008310. (End)

A370259 a(n) = (T(n,n+1) - 1)/n^3 for n >= 1, where T(n,x) is the n-th Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 2, 9, 75, 961, 16900, 380689, 10498005, 343323841, 13007560326, 560789801881, 27125634729375, 1455389462287489, 85805768251305992, 5515372218107327521, 383931652351786775721, 28778117694539885440129, 2311202255914842794592010, 198009919900727928789497641, 18027589454633803742596931571

Offset: 1

Peter Bala, Mar 11 2024

It appears that a(2*n+1) is always a square, while a(2*n) = (n + 1) * a square. See A370260 and A370261.

Paolo Xausa, Table of n, a(n) for n = 1..350

Cf. A008310, A342205, A370260, A370261.

seq( simplify( (ChebyshevT(n, n+1) - 1)/n^3 ), n = 1..20);

Array[(ChebyshevT[#, #+1]-1)/#^3 &, 20] (* Paolo Xausa, Mar 14 2024 *)

from sympy import chebyshevt
def A370259(n): return (chebyshevt(n,n+1)-1)//n**3 # Chai Wah Wu, Mar 13 2024

a(n) = Sum_{k = 1..n} (2^k)*n^(k-2)*binomial(n+k, 2*k)/(n + k) (shows that a(n) is an integer).
a(n) = (cos(n*arccos(n+1)) - 1)/n^3.
a(n) = (A342205(n) - 1)/n^3.
a(n) = ( (n + 1 + sqrt(n*(n+2)))^n + (n + 1 - sqrt(n*(n+2)))^n - 2 )/(2*n^3).

cp's OEIS Frontend

A063496 a(n) = (2*n - 1)*(8*n^2 - 8*n + 3)/3.

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A028297 Coefficients of Chebyshev polynomials of the first kind: triangle of coefficients in expansion of cos(n*x) in descending powers of cos(x).

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A001794 Negated coefficients of Chebyshev T polynomials: [x^n](-T(n+6, x)), n >= 0.

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A039991 Triangle of coefficients of cos(x)^n in polynomial for cos(nx).

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A123027 Triangle of coefficients of (1 - x)^n*U(n,-(3*x - 2)/(2*x - 2)), where U(n,x) is the n-th Chebyshev polynomial of the second kind.

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A008312 Triangle of coefficients of Chebyshev polynomials U_n(x).

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A063496 a(n) = (2n - 1)(8n^2 - 8n + 3)/3.

A123027 Triangle of coefficients of (1 - x)^nU(n,-(3x - 2)/(2*x - 2)), where U(n,x) is the n-th Chebyshev polynomial of the second kind.

A053112 Expansion of (-1 + 1/(1-9x)^9)/(81x); related to A053108.